What if your parents want to redo the bathroom? Below is the tile they would like to place in the shower. The blue and green triangles are all equilateral. What type of polygon is dark blue outlined figure? Can you determine how many degrees are in each of these figures? Can you determine how many degrees are around a point? After completing this Concept, you'll be able to apply important properties about equilateral triangles to help you solve problems like this one.

Guidance

By definition, all sides in an equilateral triangle have exactly the same length.

Investigation: Constructing an Equilateral Triangle

Tools Needed: pencil, paper, compass, ruler, protractor

1. Because all the sides of an equilateral triangle are equal, pick a length to be all the sides of the triangle. Measure this length and draw it horizontally on your paper.

2. Put the pointer of your compass on the left endpoint of the line you drew in Step 1. Open the compass to be the same width as this line. Make an arc above the line.

3. Repeat Step 2 on the right endpoint.

4. Connect each endpoint with the arc intersections to make the equilateral triangle.

Use the protractor to measure each angle of your constructed equilateral triangle. What do you notice?

From the Base Angles Theorem, the angles opposite congruent sides in an isosceles triangle are congruent. So, if all three sides of the triangle are congruent, then all of the angles are congruent or
each.

Equilateral Triangles Theorem:
All equilateral triangles are also equiangular. Also, all equiangular triangles are also equilateral.

Example A

Find the value of
.

Because this is an equilateral triangle
. Now, we have an equation, solve for
.

Example B

Find the values of
and
.

Let’s start with
. Both sides are equal, so set the two expressions equal to each other and solve for
.

For
, we need to use two
expressions because this is an isosceles triangle and that is the base angle measurement. Set all the angles equal to
and solve.

Example C

Two sides of an equilateral triangle are
units and
units. How long is each side of this triangle?

The two given sides must be equal because this is an equilateral triangle. Write and solve the equation for
.

To figure out how long each side is, plug in
for
in either of the original expressions.
. Each side is
units.

Concept Problem Revisited

Let’s focus on one tile. First, these triangles are all equilateral, so this is an equilateral hexagon (6 sided polygon). Second, we now know that every equilateral triangle is also equiangular, so every triangle within this tile has
angles. This makes our equilateral hexagon also equiangular, with each angle measuring
. Because there are 6 angles, the sum of the angles in a hexagon are
or
. Finally, the point in the center of this tile, has
angles around it. That means there are
around a point.

Vocabulary

An
isosceles triangle
is a triangle that has
at least
two congruent sides. The congruent sides of the isosceles triangle are called the
legs
. The other side is called the
base
. The angles between the base and the legs are called
base angles
and are always congruent by the
Base Angles Theorem
. The angle made by the two legs is called the
vertex angle
. An
equilateral triangle
is a triangle with three congruent sides.
Equiangular
means all angles are congruent. All equilateral triangles are equiangular.

Guided Practice

1. Find the measure of
.

2. Fill in the proof:

Given
: Equilateral
with

Prove
:
is equiangular

Statement

Reason

1.

1. Given

2.

2. Base Angles Theorem

3.

3. Base Angles Theorem

4.

4. Transitive PoC

5.
is equiangular

5.

3. True or false: All equilateral triangles are isosceles triangles.

Answers:

1. The markings show that all angles are congruent. Since all three angles must add up to
this means that each angle must equal
. Write and solve an equation:

2.

Statement

Reason

1.

1. Given

2.

2. Base Angles Theorem

3.

3. Base Angles Theorem

4.

4. Transitive PoC

5.
is equiangular

5. Definition of equiangular.

3. This statement is true. The definition of an isosceles triangle is a triangle with at least two congruent sides. Since all equilateral triangles have three congruent sides, they fit the definition of an isosceles triangle.